One of the beautiful things about radio is that there is no intermediary between sender and receiver. Unlike the &$%*@! Internet.
Thanks to golfguru for the assist while AT&T went on walkabout!
Let me give a little theory behind the Hoop Loop. A tuned loop antenna is a system. You have the loop of wire and a tuning capacitor. So the antenna functions as the inductor in a simple resonant circuit. Tuning capacitors with maximum capacitance of between 365 and 500 picofarads (pF -- pico = 10^-12) are common, or were in the recent past, and there are still plenty of them around. Most loop antennas are designed to be tuned by such a capacitor. In other words, the number of turns of wire is chosen for convenience, not magick. The physical dimensions of the loop are also chosen for convenience. Fortunately for us, it turns out that loops about 2-4 feet (3/4 to 1-1/4 m) across are not only easy to use indoors, but also work pretty well on the broadcast band.
(Unfortunately for shortwave listeners, loops like this work less well above 2-3 MHz, but we'll defer discussion of why for another time.)
The Hoop Loop also works as the inductance in a tuned circuit, but there are a few tricks along the way. It came about when I was looking for a better broadcast-band antenna to use on a Tecsun PL-380 portable radio. It uses a chip invented and manufactured in the U.S., though the radios are assembled in China. The chip has a built-in tuning capacitor that tunes the built-in ferrite-core loop antenna. The range of the internal capacitor is similar to the mechanical variable capacitors I mentioned earlier, but the upper and lower capacitance limits are farther apart. In other words, it will tune a more diverse range of antenna inductances than a mechanical variable capacitor.
Looking over the data sheet for the chip, I noticed mention of using a matching transformer to work with a loop antenna of much lower inductance. You get antennas like this with stereo receivers. They're 5-6 inches (about 150 mm) in diameter and have only a few turns of wire. They connect to the receiver with two-conductor speaker wire. Intrigued, I bought one on eBay and made a matching transformer. Sure enough, the radio still tuned the antenna, meaning I didn't have to separately peak the antenna as I tuned around the band. I knew a larger-diameter loop would give better signal strengths and deeper nulls (the main performance feature of a loop), and the Hoop Loop was the result.
The concept is simple. The tuning capacitor is looking for a certain inductance, so one winding of the transformer provides it. The other winding has fewer turns, thus stepping down the inductance to match that of the loop. When you connect the loop to the smaller winding, its inductance is reflected back to the larger winding, which you have to take into account when calculating the number of turns on each winding. But because the chip in the Tecsun radio has such a wide tuning range, getting the turns ratio in the ballpark was good enough.
I later made an arrangement using a mechanical 365-pF variable cap, but because its range is smaller and predetermined, it took a little more pencil scratching to get the inductances sorted. The advantage is that the tuning cap can be right next to the radio, rather than co-located with the antenna. I set the antenna on a little Lazy Susan turntable on top of test equipment on my bench, where it's easy to spin, but it would be inconvenient to have to keep reaching up to tune it. Also, because the feedline between the antenna is balanced and low impedance, speaker wire works just great, and the antenna could be even farther away, even outside. Then I'd want to use motors to turn and tilt it, but it's possible. Much easier to run motor drives than remote tuning via voltage-variable capacitors.
If I haven't put dan d to sleep yet, I wouldn't recommend starting with the Hoop Loop arrangement. It's going to be enough effort to couple a loop antenna to your one-tube regen, and the Hoop Loop is probably a more-complicated project than you need to take on just yet. To be frank, I'd recommend just getting a Terk antenna, or a Kaito if you like the looks, and then we can figure out how to connect it to your receiver. You see, adding a tuned circuit ahead of a one-tube regen (no isolation stage between detector and antenna) can make you crazy, as tuning the antenna may detune the radio. But first, see how well the helical vertical antenna works. The winter DX season is nearing its end, and summer static is on the way, so you'll want to spend some time logging stations and getting experience with your radio. If you're still hooked you can decide if you want to concentrate on DXing, or taking on a homebrew radio project. Putting too many variables into the equation too soon turns fun into frustration!